Integral of xarctg3xdx dx

The solution

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  1               
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 |  x*atan(3*x) dx
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0

$$\int\limits_{0}^{1} x \operatorname{atan}{\left(3 x \right)}\, dx$$

Integral(x*atan(3*x), (x, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \operatorname{atan}{\left(3 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$ .

Then $\operatorname{du}{\left(x \right)} = \frac{3}{9 x^{2} + 1}$ .

To find $v{\left(x \right)}$ :
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{3 x^{2}}{2 \left(9 x^{2} + 1\right)}\, dx = \frac{3 \int \frac{x^{2}}{9 x^{2} + 1}\, dx}{2}$
1. Rewrite the integrand:
  
  $\frac{x^{2}}{9 x^{2} + 1} = \frac{1}{9} - \frac{1}{9 \left(9 x^{2} + 1\right)}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{9}\, dx = \frac{x}{9}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{9 \left(9 x^{2} + 1\right)}\right)\, dx = - \frac{\int \frac{1}{9 x^{2} + 1}\, dx}{9}$
    So, the result is: $- \frac{\operatorname{atan}{\left(3 x \right)}}{27}$
  The result is: $\frac{x}{9} - \frac{\operatorname{atan}{\left(3 x \right)}}{27}$
So, the result is: $\frac{x}{6} - \frac{\operatorname{atan}{\left(3 x \right)}}{18}$
Add the constant of integration:

$\frac{x^{2} \operatorname{atan}{\left(3 x \right)}}{2} - \frac{x}{6} + \frac{\operatorname{atan}{\left(3 x \right)}}{18}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \operatorname{atan}{\left(3 x \right)}}{2} - \frac{x}{6} + \frac{\operatorname{atan}{\left(3 x \right)}}{18}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                      2          
 |                      x   atan(3*x)   x *atan(3*x)
 | x*atan(3*x) dx = C - - + --------- + ------------
 |                      6       18           2      
/

$$\int x \operatorname{atan}{\left(3 x \right)}\, dx = C + \frac{x^{2} \operatorname{atan}{\left(3 x \right)}}{2} - \frac{x}{6} + \frac{\operatorname{atan}{\left(3 x \right)}}{18}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   5*atan(3)
- - + ---------
  6       9

$$- \frac{1}{6} + \frac{5 \operatorname{atan}{\left(3 \right)}}{9}$$

  1   5*atan(3)
- - + ---------
  6       9

$$- \frac{1}{6} + \frac{5 \operatorname{atan}{\left(3 \right)}}{9}$$

-1/6 + 5*atan(3)/9

Numerical answer [src]

0.527247651332364

0.527247651332364

Use the examples entering the upper and lower limits of integration.

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Integral of xarctg3xdx dx

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The solution